A random sample of the amount paid (in dollars) for taxi fare from downtown to the airport was obtained. (Give your answers correct to two decimal places.)
16 17 22 13 21 23 22 18 13 22 19 20 19 22 16
(a) Use the data to find a point estimate for the mean.
$ 18.87 .
(b) Use the data to find a point estimate for the variance.
$ I have a problem getting variance
(c) Use the data to find a point estimate for the standard deviation.
$ I have a problem getting sd.
I would do c) first
probably the most common method to find the SD
is to take differences between each of the given data values and the mean.
This will give you 15 differences, some positive , some negative
make a new list of the square of each of these differences.
Add up these squares and divide that sum by 14
(one less than the number of data values, you might want to check with your text or notes on this one, some divide by n itself, the number of data values)
Take the square root of that division answer and that is your SD
b) variance = (SD)^2
here is a page illustrating SD vs variance
http://stats.stackexchange.com/questions/35123/whats-the-difference-between-variance-and-standard-deviation
To find a point estimate for the variance, we can use the formula:
Variance = (1 / n) * Σ(x - μ)^2
where n is the sample size, x is each individual data point, and μ is the mean.
First, let's find the mean (point estimate for the mean):
Mean = (16 + 17 + 22 + 13 + 21 + 23 + 22 + 18 + 13 + 22 + 19 + 20 + 19 + 22 + 16) / 15
= 306 / 15
= 20.40
Now, let's calculate the variance using the formula:
Variance = (1 / 15) * ((16 - 20.40)^2 + (17 - 20.40)^2 + ... + (16 - 20.40)^2)
Variance = (1 / 15) * (157.12 + 124.36 + ... + 157.12)
Variance = (1 / 15) * 1923.2
Variance ≈ 128.21
Therefore, the point estimate for the variance is approximately $128.21.
To find a point estimate for the standard deviation, we can take the square root of the point estimate for the variance:
Standard Deviation = √(128.21)
Standard Deviation ≈ 11.32
Therefore, the point estimate for the standard deviation is approximately $11.32.
To calculate the point estimate for the mean, variance, and standard deviation, you can follow these steps:
(a) Point estimate for the mean:
To find the point estimate for the mean, simply average the values in the sample set.
In this case, sum up all the values and divide by the number of values in the sample:
16 + 17 + 22 + 13 + 21 + 23 + 22 + 18 + 13 + 22 + 19 + 20 + 19 + 22 + 16 = 291.
Now divide the sum by the number of values, which is 15:
291 / 15 = 19.4.
So, the point estimate for the mean is $19.40.
(b) Point estimate for the variance:
To find the point estimate for the variance, you need to calculate the squared differences between each value and the mean, and then average those squared differences.
First, calculate the squared differences for each value. Subtract the mean from each value and square the result:
(16 - 19.4)^2 = 12.96
(17 - 19.4)^2 = 5.76
(22 - 19.4)^2 = 6.76
(13 - 19.4)^2 = 40.96
(21 - 19.4)^2 = 2.56
(23 - 19.4)^2 = 13.84
(22 - 19.4)^2 = 6.76
(18 - 19.4)^2 = 2.44
(13 - 19.4)^2 = 40.96
(22 - 19.4)^2 = 6.76
(19 - 19.4)^2 = 0.16
(20 - 19.4)^2 = 0.36
(19 - 19.4)^2 = 0.16
(22 - 19.4)^2 = 6.76
(16 - 19.4)^2 = 12.96
Next, average these squared differences by summing them up and dividing by the number of values in the sample - 1 (since we're estimating the variance of the population from a sample):
(12.96 + 5.76 + 6.76 + 40.96 + 2.56 + 13.84 + 6.76 + 2.44 + 40.96 + 6.76 + 0.16 + 0.36 + 0.16 + 6.76 + 12.96) / 14 ≈ 10.53.
So, the point estimate for the variance is approximately $10.53.
(c) Point estimate for the standard deviation:
To find the point estimate for the standard deviation, take the square root of the estimated variance.
Using the previous calculated variance of $10.53, take the square root:
√10.53 ≈ 3.24.
So, the point estimate for the standard deviation is approximately $3.24.